Iterative Unbiased Converted Measurement Kalman Filter for Target Tracking
Da Li
1,2
,
Xiangyu Zou
1,2
,
Ruifang Li
1,2
1
School of Information Engineering, Wuhan University of Technology, Wuhan, Hubei, China, 430070.
2
Key Laboratory of Fiber Optic Sensing Technology and Information Processing , Ministry of Education, Wuhan
University of Technology, Wuhan, Hubei, China, 430070.
frankli26@whut.edu.cn, variance@whut.edu.cn, liruifang@whut.edu.cn
Abstract—It is important to develop a robust and fast tracking
algorithm for modern tracking system because the target moves
more and more fast. Therefore, a novel algorithm named the
iterative unbiased converted measurement Kalman filter
(IUCMKF) is proposed based on the analysis and comparison of
conventional target tracking methods. The new algorithm is
developed from unbiased converted measurement Kalman
filter, but it can obtain more accurate state and covariance
estimation. Compared with the existing target tracking
approaches, the proposed method has potential advantages in
tracking accuracy. The correctness as well as validity of the
algorithm is demonstrated through numerical simulation
results.
Keywords-iteration method; kalman filter; target tracking;
unbiased converted measurement
I. INTRODUCTION
The basic filtering solution to the state estimation problem
for a discrete-time dynamic system can be described as a two-
stage recursive process of update and prediction.
Usually the target motion can be best modeled in Cartesian
coordinates, but the measurements are often obtained in polar
coordinates. For solving this mismatch, there are two common
ways. One method is to use the extended Kalman filter (EKF).
The EKF incorporates the original measurements in a
nonlinear fashion into the target state estimation, within a
mixed coordinate filter, and its basic idea is to first linearly
approximate the nonlinear function, then utilize the basic
equation of the linear system Kalman filter to achieve the state
estimation. Under EKF framework, the nonlinear function is
linearized at the state estimate using Taylor series expansion.
This method is suboptimal or sometimes lead to divergence of
the filter [1][2], which relies on the accuracy of
approximations. Another method is to use a linear Kalman
filter by converting the raw measurements to a reference
Cartesian coordinate frame. This method is so-called
converted measurement Kalman filter (CMKF). As a result,
the elements of errors in the converted measurements are
correlated.
In the target tracking system, the state equation of the
target is usually established in the Cartesian coordinate
system, and the measurement is generally measured in the
polar (spherical) coordinate system by the corresponding
sensor. To solve some of the problems existing in the EKF,
Bar-Shalom [3] et al proposed a CMKF algorithm on a two-
dimensional plane. The simulation results show that the
CMKF is better than the EKF in the tracking of nonlinear
systems. However, the traditional CMKF algorithm will
produce some error in the conversion measurement, resulting
in tracking inconsistencies. Therefore, the improved CMKF
algorithms are proposed including the debiased conversion
measurement Kalman filter (DCMKF) [4][5]and the unbiased
conversion measurement Kalman filter (UCMKF) [6].
In order to optimize the performance of UCMKF method,
an iterative unbiased converted measurement Kalman filter
(IUCMKF) method is proposed based on the iterative idea of
IEKF. The new method can obtain more accurate target state
and covariance estimation. Compared with traditional target
tracking methods (such as CMKF and UCMKF), the proposed
method has potential advantages in tracking accuracy.
Meanwhile, the new method also has faster convergence rate
and lower mean square root error and mean error.
II. D
EVELOPMENT OF THE IUCMKF
A. CMKF and UCMKF
The target motion model is expressed in the Cartesian
coordinate system by a linear discrete-time difference
equation
1kkkk
xFxv
+
=+
(1)
and the measurement equation is given as
()
kkk
hx n=+
(2)
where
k
x
is the Cartesian state vector,
k
is the polar
measurement vector,
k
v
and
k
n
denote the process and the
measurement noise vector with corresponding covariance
matrices and , respectively. The measurement noises
are assumed to be mutually uncorrelated and zero-mean
white Gaussian.
It is assumed that the process noise and the measurement
noise are mutually independent of each other and of the initial
state
0
x
. All matrices describing the system and the
covariance matrices are assumed to be known.
To be simple and clear the main idea and derivations are
given for the 2D case, and the extension to the 3D case is just
around the corner. For 2D case, the measured range and
azimuth can be defined as
;
mrm
rr
θ
θθν
=+ =+
(3)
where are the true range and azimuth angle of the target,
and are components of
k
n
with variances and
respectively.
k
Q
k
,r
θ
r
θ
2
r
σ
2
θ
σ
2017 10th International Symposium on Computational Intelligence and Design
2473-3547/17 $31.00 © 2017 IEEE
DOI 10.1109/ISCID.2017.52
341
2017 10th International Symposium on Computational Intelligence and Design
2473-3547/17 $31.00 © 2017 IEEE
DOI 10.1109/ISCID.2017.52
342